Here we discuss the 1D algorithm to estimate \(\mathrm {span}({\mathbf {\Gamma }})\) . The 1D algorithm was first proposed in [1]} for multivariate regression envelope model. It obtains an estimate of \({\mathbf {\Gamma }}\)  column-wisely. Specifically, assuming that the envelope dimension \(u\)  is given, we use the following 1D algorithm to estimate \({\mathbf {\Gamma }}\)  under the mixed effects model.
